Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

Abstract

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single-layer homogeneous fluid with a constant-pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

Mathematics Subject Classification

Notes

Acknowledgements

We thank R. Colombo and M. Garavello for discussions and useful comments on the theory of quasilinear PDEs. R.C. and G.O. thank S.L. Gavrilyuk for bringing to their attention, after a first draft of this work had been submitted for publication, reference Ovsyannikov (1979) during the Summer school “Dispersive hydrodynamics and oceanography: from experiments to theory” 27 August - 1 September 2017, Les Houches (France). Support by NSF Grants DMS-0908423, DMS-1009750, DMS-1517879, RTG DMS-0943851, CMG ARC-1025523, ONR Grants N00014-18-1-2490, DURIP N00014-12-1-0749, ERC Grant H2020-MSCA-RISE-2017 PROJECT No. 778010 IPaDEGAN, and the auspices of the GNFM Section of INdAM are all gratefully acknowledged. R.C. and M.P. thank the Dipartimento di Matematica e Applicazioni of Università Milano-Bicocca for its hospitality. In addition, the Istituto Nazionale di Alta Matematica (INdAM) and the Institute for Computational and Experimental Research in Mathematics (ICERM), supported by the National Science Foundation under Grant DMS-1439786, are gratefully acknowledged for hosting the visits of R.C. (INdAM, Summer 16) and R.C. & C.T. (ICERM, Spring 17) while some of this work was carried out. Last, but not least, we would like to thank the anonymous referees whose attentive reading of the manuscript greatly helped improve it.

Appendix A: The Limiting Case of Air–Water System

gives rise for \(r=0\) to the Boussinesq two-layer model smoothly in r (i.e., the Hamiltonian is continuous in r at \(r=0\)). The opposite limit \(r\rightarrow 1\) (that is, the air–water system) is more subtle. In fact, as long as the interface does not get into contact with the top boundary \(\eta =1\) of the channel, expression (4.6) coincides with the classical Airy Hamiltonian density

For every fixed value \(\rho _r\ll 1\), if the interface gets sufficiently close to the upper lid, the correction term of the Hamiltonian density can no longer be assumed to be smaller than the leading-order term. In other words, an asymptotic expansion in the small parameter \(\rho _r=1-r\) fails in a neighborhood of \(\eta =1\).

Let us now consider the hyperbolicity region of the Hamiltonian system generated by (4.6), i.e., the domain in the \((\sigma ,\eta )\)-plane where the characteristic velocities \(\lambda _\pm \) are real and distinct (see Fig. 21). It is easy to show that its boundary (the so-called transition line) is given by \(H_{\sigma \sigma }=0\) and \(H_{\eta \eta }=0\). The first equation gives the physical boundaries of the system: \(\eta =0\) and \(\eta =1\). The second equation is

When \(r\rightarrow 1\), the hyperbolic domain fills the whole configuration space \(\mathbb {R}\times [0,1]\). However, the transition line \(H_{\eta \eta }=0\) does not merge smoothly with the horizontal line \(\eta =1\), since the last derivative in (A.5) tends to \({\mp }{2}/{3}\). As a consequence, if the distance of some point of the interface is smaller than the infinitesimal \(1-r\), the system will experience a hyperbolic-elliptic transition if the weighted vorticity \(\sigma \) is larger than an infinitesimal of the same order.

is not differentiable at \(x=0\), and the critical value of the field \(\eta \) is \(\eta _c=0\). Hence, the shock forms at the boundary of the domain, and one can check that \(x_c=-2/3\) and \(t_c=1/3\). We can conclude that for this example the smoothing does not introduce qualitatively new phenomena on the position of the shock formation with respect to the non-smooth cases considered in Sect. 3.1 for simple waves solutions of the Airy equations.